Inclined Plane when subjected to alternations of Temperature. 101 



the dilatation of the greater length of the plate X B favoured by- 

 its weight that the extremity B is displaced down the plane when 

 the temperature is raised; whilst it is by the contraction of the 

 less length X L B against its weight that it is displaced up the 

 plane when the temperature is lowered. The extremity B is 

 therefore more displaced down the plane by a given raising of 

 the temperature than it is displaced up it by a corresponding- 

 lowering. On the whole, therefore, the extremity B is made to 

 descend the plane by a given alternation of temperature. It is 

 by the dilatation of the less length X A that the extremity A is 

 displaced up the plane, aud by the contraction of the greater 

 length Xj Aj that it is displaced down the plane. It is there- 

 fore less displaced up by dilatation than it is down by contrac- 

 tion, and on the whole it descends by a given alternation of tem- 

 perature. Both the extremities A and B of the plate are there- 

 fore made to descend, when it is subjected to a given elevation 

 and then to a corresponding depression of its temperature; that 

 is, the whole plate is made to descend. 



It is the object of the following paper to discuss the mathe- 

 matical conditions of this descent with a view to its application to 

 the theory of the descent of glaciers. Formula (22) is the ma- 

 thematical expression of the result. 



Two principal cases arise in this discussion. An increase 

 of temperature has been supposed to be communicated to an 

 element at X such as would be sufficient, if the plate were di- 

 vided at that point, to push XA up the plane and XB down. 

 This determines the increment of temperature with reference to 

 the length of the plate; and so of the corresponding decre- 

 ment, which must be sufficient to pull Xj Aj down and Xj Bj up. 

 The first case is that in which the alternation of temperature to 

 which the plate is subjected is equal to, or greater than, this. 

 In this case the plate descends. The second case is that in 

 which the alternation of temperature is less than this. The 

 thrust of dilatation produced by the given increment of tempe- 

 rature of an element at X which is sufficient to push X A up 

 not being sufficient to push X B down, let x B (fig. 2) be a part 

 of the plate which it would be just sufficient to push down, and 

 let the whole plate receive this increment of temperature. The 

 parts X A and x B will then be dilated to A, and B„ but X x 

 will remain undilated. Let now Xj be a point in the plate at 

 which, if an element experience a corresponding decrease of tem- 

 perature, the strain of its contraction would be sufficient to pull 

 X Aj down but not Xj B T up ; and let x Y B x be the part of the 

 plate that it would just pull up. If, then, the whole plate expe- 

 rience this decrement of temperature, Xj A x will be contracted 

 to Xj A 2 , and x x B t to x } B 2; but X l x l will remain uncontracted. 



